 Uniform derivative convergence with an anchored value

If continuously differentiable functions have uniformly convergent derivatives on an interval and their values converge at one point, the functions converge locally uniformly to a continuously differentiable limit whose derivative is the derivative limit. The proof integrates from the anchored point using the fundamental theorem of calculus. Uniform convergence of the functions on the entire interval follows when the interval is bounded; it need not follow on an unbounded interval.